### All GRE Subject Test: Chemistry Resources

## Example Questions

### Example Question #11 : Random Variables

Robert's work schedule for next week will be released today. Robert will work either 45, 40, 25, or 12 hours. The probabilities for each possibility are listed below:

45 hours: 0.3

40 hours: 0.2

25 hours: 0.4

12 hours: 0.1

What is the standard deviation of the possible outcomes?

**Possible Answers:**

**Correct answer:**

There are four steps to finding the standard deviation of random variables. First, calculate the mean of the random variables. Second, for each value in the group (45, 40, 25, and 12), subtract the mean from each and multiply the result by the probability of that outcome occurring. Third, add the four results together. Fourth, find the square root of the result.

### Example Question #1 : Variance And Standard Deviation

We have two independent, normally distributed random variables and such that has mean and variance and has mean and variance . What is the probability distribution of the difference of the random variables, ?

**Possible Answers:**

Normal distribution with mean and variance .

Normal distribution with mean and variance .

Normal distribution with mean and variance .

Normal distribution with mean and variance .

**Correct answer:**

Normal distribution with mean and variance .

The mean for any set of random variables is additive in the sense that

The difference is also additive, so we have

This means the mean of is .

The variance is additive when the random variables are independent, which they are in this case. But it's additive in the sense that for any real numbers (even when negative), we have

.

So for this difference, we have

.

So the mean and variance are and , respectively. In addition to that, is normally distributed because the sum or difference of any set of independent normal random variables is also normally distributed.

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